We use p-adic analytic methods to analyze automorphisms of smooth projective varieties. We prove a version of the dynamical Mordel-Lang conjecture for arbitrary subschemes of a variety. We apply this result to (1) classify automorphisms of X for which there exists a divisor D whose intersection with its iterates are not dense in D, and (2) show that various properties of Aut(X) (for example finiteness of its component group) are not altered by blowups in high codimension. This is joint work with John Lesieutre.